For a module L the formal Dirichlet series ζL(s) = ∑n ≥ 1ann-s is defined whenever the number an of submodules of L with index n is finite for each positive integer n. For a ring R and a finite association scheme (X,S) we denote the adjacency algebra of (X,S) over R by RS. In this talk we aim to compute ζZS(s) where ZS is regarded as a ZS-module under the assumption that |X| is prime or |S|=2.

This is a joint work with Cai-Heng Li. Let \(\mathcal{MF}\) denote the set of positive integers n such that each transitive action of degree n is multiplicity-free, and \(\mathcal{PQ}\) denote the set of \(n\in \mathbb{N}\) such that n=pq for some primes p, q with \(p[latex]\{pq\in \mathcal{PQ}\mid (p,q-2)=p, q\mbox{ is a Fermat prime}\}\) and